Electrical Engineering 2024 Paper I 50 marks Derive

Q6

(a) (i) An AM signal s(t) = A_c[1 + k_a m(t)]cos(2πf_c t) is applied to the system shown in the figure. Show that the message signal m(t) can be obtained from the square-rooter output v₃(t): Assume that |k_a m(t)| < 1 for all t, the message signal m(t) is limited to the interval −ω ≤ f ≤ ω, and the carrier frequency f_c > 2ω. (10 marks) (ii) A narrow band FM signal is approximately given as $$s(t) \approx A_c \cos(2\pi f_c t) - \beta A_c \sin(2\pi f_c t)\sin(2\pi f_m t)$$ Determine the envelope of this modulated signal. Also determine the ratio of the maximum to the minimum value of this envelope. Plot this ratio versus β, with β restricted to the interval 0 ≤ β ≤ 0·4. Also determine the average power of the narrow band FM signal, expressed as a percentage of the average power of the unmodulated carrier wave. (10 marks) (b) (i) Explain why PWM inverters are preferred over square wave inverters. Further, draw the harmonic spectrum to highlight the differences in unipolar and bipolar PWM techniques. (10 marks) (ii) A single-phase, full-bridge inverter has DC-link voltage $V_{DC} = 400$ V, and the fundamental frequency of 50 Hz. Find the r.m.s. value of the voltages of the fundamental and next two prominent harmonics for the following cases: (1) Square wave mode (2) Voltage cancellation mode with α = 20° (10 marks) (c) A 50 hp, 440 V, 50 Hz, star-connected, three-phase induction motor has a starting torque of 75% and maximum torque of 250% of the full-load torque. Find the following: (i) Slip at which maximum torque occurs (ii) Slip at full-load torque (10 marks)

हिंदी में प्रश्न पढ़ें

(a) (i) एक AM संकेत s(t) = A_c[1 + k_a m(t)]cos(2πf_c t) चित्र में प्रदर्शित तंत्र पर अनुप्रयुक्त है। दिखाइए कि संदेश संकेत m(t) को वर्गमूलक निर्गत v₃(t) से प्राप्त किया जा सकता है: मान लीजिए कि t के सभी मानों के लिए |k_a m(t)| < 1 है, संदेश संकेत m(t) अन्तराल −ω ≤ f ≤ ω में सीमित है और वाहक आवृत्ति f_c > 2ω है। (10 अंक) (ii) एक संकीर्ण बैंड FM संकेत लगभग निम्न द्वारा निर्धारित है: $$s(t) \approx A_c \cos(2\pi f_c t) - \beta A_c \sin(2\pi f_c t)\sin(2\pi f_m t)$$ इस मॉडुलित संकेत का आवरण (एनवेलप) ज्ञात कीजिए। इस आवरण के अधिकतम व न्यूनतम मान का अनुपात भी ज्ञात कीजिए। इस अनुपात का β के सापेक्ष आलेख कीजिए, जबकि β अंतराल 0 ≤ β ≤ 0·4 में सीमित है। साथ ही संकीर्ण बैंड FM संकेत की औसत शक्ति का मान, अमॉडुलित वाहक तरंग की औसत शक्ति के प्रतिशत के रूप में व्यक्त कीजिए। (10 अंक) (b) (i) PWM प्रतिपथों (इन्वर्टरों) को वर्ग तरंग प्रतिपथों की अपेक्षा क्यों ज्यादा पसंद किया जाता है, व्याख्या कीजिए। एक-ध्रुवीय एवं द्वि-ध्रुवीय PWM तकनीकों में अंतर को उजागर करने के लिए सनादी स्पेक्ट्रम को आरेखित कीजिए। (10 अंक) (ii) एक एकल कला वाला पूर्ण-ब्रिज प्रतिपथ की DC-लिंक वोल्टता $V_{DC} = 400$ V एवं मूल आवृत्ति 50 Hz है। मूल एवं आगे के दो प्रमुख सनादी की वोल्टता के r.m.s. मान को निम्नलिखित प्रकरणों में ज्ञात कीजिए: (1) वर्ग तरंग विधा (2) वोल्टता निरस्तीकरण विधा, जब α = 20° है (10 अंक) (c) एक 50 hp, 440 V, 50 Hz, तारा-संयोजित, त्रिकला प्रेरण मोटर का आरंभिक बल-आघूर्ण, पूर्ण-भार बल-आघूर्ण का 75% और अधिकतम बल-आघूर्ण, पूर्ण-भार बल-आघूर्ण का 250% है। निम्नलिखित ज्ञात कीजिए: (i) सर्पण, जिस पर बल-आघूर्ण अधिकतम होता है (ii) पूर्ण-भार बल-आघूर्ण पर सर्पण (10 अंक)

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Directive word: Derive

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Approach

Begin with a brief introduction acknowledging the three distinct domains: AM/FM demodulation, PWM inverter analysis, and induction motor characteristics. Allocate approximately 25% time to part (a) covering AM envelope detection and FM envelope/power calculations; 25% to part (b) on PWM advantages and harmonic analysis with spectra; 25% to part (c) on torque-slip characteristics; reserve 25% for diagrams, numerical verification, and conclusion. For (a)(i), derive the square-rooter output step-by-step; for (a)(ii), use trigonometric identities for envelope extraction; for (b), contrast unipolar/bipolar PWM spectra; for (c), apply the torque-slip equation T ∝ s/(r₂² + (sx₂)²).

Key points expected

  • (a)(i) Derivation showing v₃(t) = A_c√[1+k_a m(t)] through squaring, filtering, and square-root operations with proper justification of LPF cutoff selection (f_c > 2ω)
  • (a)(ii) Envelope derivation using A(t) = A_c√[1 + β²sin²(2πf_m t)], ratio (1+β)/(1-β) for small β, correct plot of ratio vs β (0 to 0.4), and power calculation showing ≈(1+β²/4)×100%
  • (b)(i) PWM advantages: reduced harmonic distortion, adjustable output voltage via modulation index, better THD; clear harmonic spectrum comparison showing unipolar eliminates even harmonics and carrier multiples while bipolar has harmonics at mf_c ± nf_o
  • (b)(ii) Square wave: V₁ = 0.9V_DC = 360V, V₃ = 120V, V₅ = 72V; Voltage cancellation: V₁ = (4V_DC/π)cosα = 428.5V, correct harmonic elimination pattern with α = 20°
  • (c) Using T_max/T_fl = 2.5 and T_st/T_fl = 0.75 with torque-slip relation: slip at T_max s_max = r₂/x₂ = 0.183, full-load slip s_fl = 0.037 (or 0.163 if using approximate method), showing both exact and approximate solutions
  • Proper use of Thevenin equivalent or approximate equivalent circuit for induction motor torque calculations with clear assumption statements

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies envelope detector operation in (a)(i), distinguishes narrow-band FM approximations in (a)(ii), explains PWM switching strategies with modulation index significance in (b), and applies exact torque-slip relationship T = (3V₁²r₂/s)/[ω_s((r₁+r₂/s)²+(x₁+x₂)²)] or its simplified form in (c)Minor conceptual errors such as incorrect filter cutoff justification in (a)(i), confusing wideband/narrowband FM conditions, or using approximate torque formula without stating assumptions in (c)Fundamental misconceptions like treating AM signal as FM, confusing unipolar/bipolar PWM characteristics, or applying DC motor torque equations to induction motor
Numerical accuracy20%10Precise calculations: envelope ratio (1+β)/(1-β) ≈ 1.816 at β=0.3, power ratio 100(1+β²/4)%, square wave harmonics V_n = 4V_DC/(nπ), voltage cancellation with correct α application, motor slips s_max = 0.183 and s_fl = 0.037 with proper unit handlingCorrect method but arithmetic errors (e.g., factor of √2 confusion in RMS, incorrect angle conversion, or rounding errors beyond acceptable limits)Order-of-magnitude errors, incorrect formula substitution, or missing critical numerical steps leading to unrealistic values (e.g., slip > 1 or negative power)
Diagram quality15%7.5Clear labeled block diagram for (a)(i) system (multiplier, LPF, square-rooter), properly scaled plot of envelope ratio vs β (0-0.4) with axes labeled, distinct harmonic spectra for unipolar (odd harmonics of f_o, sidebands around mf_c) vs bipolar PWM (harmonics at mf_c ± nf_o), and torque-slip characteristic curve for (c)Diagrams present but poorly labeled, missing critical features (e.g., no distinction between unipolar/bipolar spectra), or hand-drawn appearance without proper scalingMissing essential diagrams, confusing sketches, or incorrect representation of spectral content (e.g., showing even harmonics in unipolar PWM)
Step-by-step derivation25%12.5Complete mathematical rigor: (a)(i) shows v₁ = s²(t), v₂ after LPF = ½A_c²[1+k_a m(t)]², then v₃ = A_c[1+k_a m(t)]; (a)(ii) uses cos(A)sin(B) identity, combines terms, applies √(a²+b²) for envelope; (b)(ii) derives Fourier coefficients; (c) establishes T_max/T_st relationship to find s_max and s_fl with clear algebraic stepsCorrect approach but skips intermediate steps, assumes results without proof, or uses 'it can be shown that' without justification; missing justification for small-angle approximationsCircular reasoning, incorrect mathematical operations (e.g., treating √(a+b) = √a + √b), or purely descriptive without any derivation where required
Practical interpretation20%10Connects theory to practice: explains why |k_a m(t)| < 1 prevents envelope distortion and phase reversal in AM, discusses β < 0.3 for valid narrowband FM approximation, relates PWM advantages to Indian grid standards (IEEE 519/IS 6160 harmonic limits), and interprets motor slips for design implications (high starting torque applications like textile mills, crane hoists)Generic statements without specific application context, or mentions practical relevance without elaborating on Indian/industrial relevanceNo practical context provided, or incorrect practical claims (e.g., suggesting square wave inverters for sensitive loads, or operating motor continuously at s_max)

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